Porous electrode models of Li-ion batteries often use approximations to eliminate the time consuming calculation in the second dimension r for the solid phase diffusion. These methods include the Duhamel’s superposition method,1 diffusion length method,3 the polynomial approximation method,4 the pseudo steady state (PSS) approach by Liu5 and the penetration depth analysis and mixed order finite element approach.6
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Each of the above listed methods has its own advantages and disadvantages when used in Li-ion battery models. The following section gives a brief summary about each of the methods and discusses their merits and usability.
2.2.1. Duhamel’s Superposition method
The Duhamel’s superposition method1 is a robust method available for representing the solid phase diffusion for constant diffusivities. This method relates the solution of a boundary value problem with time dependent boundary conditions to the solution of a similar problem with time-independent boundary conditions by means of a simple relation. More information about the method and equations are presented elsewhere.1, 2
Duhamel’s superposition method is a robust method and is valid for any kind of operating condition, like high rates of discharge, pulse power, etc. One of the major drawbacks of this method is that, it cannot be used in DASSL like solvers which do not accept equations discretized in time and might as well be time consuming for very stiff problems depending on the time steps taken. In addition, it cannot be used for nonlinear diffusivities.
2.2.2. Diffusion Length Method
The diffusion length method’s approach3 is based on a parabolic profile approximation for the solid phase. The diffusion length method predicts that the surface concentration and volume averaged concentration inside a particle are linearly dependent on each other, which should be valid only after the diffusion layer builds up to its steady state. Therefore, the method is inadequate at short times or under dynamic operations, such as pulse or current interrupt operations. The prediction based on the diffusion length method is inadequate at short times and very efficient at long times and low rates.
61 2.2.3. Polynomial Approximation
The polynomial approximation method by Subramanian et al.4 is based on parabolic profile approximation and volume averaging of the solid phase diffusion equation. This high order polynomial method uses a different approach from the diffusion length method to improve the solution accuracy at short times. The diffusion length method uses the empirical exponential term in the equation and determines the multiplier value by matching surface concentration profiles to the exact solutions. The high order polynomial method uses a higher order polynomial for the concentration profile in the derivation, and one could derive new sets of equations with an even higher order polynomial model, if needed, following the same procedures discussed in the papers.4, 7
This method is very efficient at long time ranges, and for low/ medium rates, and is ideal for adaptive solvers for pseudo-2D models. However, it is inaccurate at short times and for high rates/pulses and hence would not be a suitable method for implementing in models for HEVs and other high rate applications.
2.2.4. Pseudo Steady State Method
The Pseudo Steady State (PSS) approach by Liu5 is very robust and by having enough number of equations, this approach can cover the entire spectrum of high/low rates, pulses, etc.
This is a form of a finite integral transform technique to eliminate the independent spatial variable r from the solid phase diffusion equation. For diffusion problems with a time dependent pore wall flux jn appears in the boundary condition and in the calculation of coefficients.
However, this method involves terms/coefficients which blow up when the number of terms increases adding numerical difficulties for simulation. More details are given in the Results and
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Discussion section when this method is compared with our proposed approach implemented in this work.
2.2.5. Penetration Depth Method
The penetration depth analysis approach was used earlier with empirically fits to numerical solution for penetration depth near the surface of the particle. The advantage of this method is that it is very accurate at short times/pulses and more accurate and efficient Penetration depth solutions can be directly obtained from the partial differential equation as discussed elsewhere.6
The drawback with this approach being the need to be reinitialized every time, and does not give a good match for varyingδ . Though this method is very accurate and efficient at short times, it is not ideal for adaptive solvers in a pseudo-2D model (increases stiffness).
2.2.6. Finite Element Method
While the governing equation1 describes solid phase concentration along the radius of each spherical particle of active material, the macroscopic model requires only the concentration at the surface, cs (x, t), as a function of the time history of local reaction current density, j(t). The PDE, is transformed from spherical to planar coordinates using and discretized in the r-direction with N suitably chosen linear elements. (They used five elements with node points placed at {0.7, 0.91, 0.97, 0.99, 1.0} × Rs.) Transformed back to spherical coordinates, the discretized system is represented as ODEs in state space form and then solved.6
The finite element node sizes were probably obtained using trial and error and perhaps may not be optimal at long times or different operating conditions. The following section describes two new methods that can be used for solid phase diffusion approximation and explains the derivation of the same.
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